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I-priors and interactions

This contains the R code for our paper.

Wicher Bergsma, Haziq Jamil (2023). Additive interaction modelling using I-priors.

The key documents are as follows.

  1. 02-sims.R and 03-sims_analysis.R contain the code for the simulations study detailed in Section 5 of the manuscript.
  2. 01-cow.R contains the code for the functional response model in the application section (Section 6) of the manuscript.

In this README, the main summary of findings are presented. Full details of the simulation results are found in the pdf files simres_corr.pdf and simres_uncorr.pdf.

Please install the developmental version of the {iprior} package.

Simulation study

Data pairs $(y_i,x_i)$, where $x_i\in\mathbb R^3$ for $i=1,\dots,n$, were simulated according to the following model

$$ y_i = \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i1}x_{i2} + \beta_5 x_{i1}x_{i3} + \beta_6 x_{i2}x_{i3} + \beta_7 x_{i1}x_{i2}x_{i3} + \epsilon_i $$

where $\epsilon_i\sim N(0,\sigma^2)$ such that $\text{Corr}(x_{ij},x_{ik})=\rho$, for $j\neq k$. The simulation settings were $n=100$, $\sigma=3$, and $\rho\in\{0,0.5\}$. The coefficients were varied according to the table below

x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 code
1 1 0 0 0.0 0.0 0.0 0.00 1000000
2 1 1 0 0.0 0.0 0.0 0.00 1100000
3 1 1 0 0.5 0.0 0.0 0.00 1101000
4 1 1 1 0.0 0.0 0.0 0.00 1110000
5 1 1 1 0.0 0.5 0.0 0.00 1110100
6 1 1 1 0.5 0.5 0.0 0.00 1111100
7 1 1 1 0.5 0.5 0.5 0.00 1111110
8 1 1 1 0.5 0.5 0.5 0.25 1111111

For each set of true values of the coefficients, the four methods proposed the likeliest model to have generated the data set, from a search of hierarchically nested interaction models. This was replicated a total of $B=10,000$ times for each true value set.

The results below show proportion of times that each method selected the true model (higher is better). The geometric mean was used as a summary measure of the simulation runs.

Uncorrelated errors

mod iprior lasso spikeslab gprior
1 1000000 0.69 0.21 0.56 0.45
2 1100000 0.55 0.33 0.32 0.29
3 1101000 0.52 0.11 0.02 0.04
4 1110000 0.33 0.37 0.18 0.13
5 1110100 0.32 0.15 0.01 0.01
6 1111100 0.26 0.09 0.00 0.00
7 1111110 0.16 0.08 0.00 0.00
8 1111111 0.19 0.06 0.00 0.98

Correlated errors

mod iprior lasso spikeslab gprior
1 1000000 0.64 0.20 0.55 0.46
2 1100000 0.54 0.43 0.11 0.27
3 1101000 0.48 0.09 0.01 0.09
4 1110000 0.43 0.52 0.01 0.16
5 1110100 0.31 0.10 0.00 0.07
6 1111100 0.27 0.10 0.00 0.00
7 1111110 0.18 0.13 0.00 0.00
8 1111111 0.43 0.16 0.00 0.78

Summary of results

Functional response model

The model fitted was an I-prior model with ANOVA kernel (Pearson & fBm). The results are tabulated below.

Fixed hurst = 0.5

model formula loglik error lambda psi hurst k AIC BIC
1 time -2789.23 16.25 0.837 0.00375 0.5 2 5582.46 5591.45
2 group * time -2789.20 16.24 0.019,-0.836 0.00375 0.5 3 5584.40 5597.88
3 id * time -2295.16 2.89 -0.203,-0.088 0.07384 0.5 3 4596.33 4609.81
4 (group + id) * time -2270.85 2.62 -1.019,-0.187,-0.085 0.08711 0.5 4 4549.70 4567.67
5 group * id * time -2249.00 3.09 -1.057,4.918,0.047 0.06538 0.5 4 4506.00 4523.97

Fixed hurst = 0.3

model formula loglik error lambda psi hurst k AIC BIC
1 time -2792.78 16.22 4.465 0.00375 0.3 2 5589.56 5598.54
2 group * time -2792.73 16.20 0.03,-4.462 0.00376 0.3 3 5591.47 5604.94
3 id * time -2266.39 1.62 -0.163,-0.381 0.13445 0.3 3 4538.79 4552.26
4 (group + id) * time -2242.30 1.44 0.708,-0.152,-0.36 0.15896 0.3 4 4492.60 4510.57
5 group * id * time -2238.78 2.16 -1.184,-1.265,0.248 0.09450 0.3 4 4485.56 4503.53

Estimated hurst value

model formula loglik error lambda psi hurst k AIC BIC
1 time -2788.77 16.28 0.347 0.00374 0.62 3 5583.53 5597.01
2 group * time -2788.75 16.27 0.013,-0.35 0.00375 0.61 4 5585.49 5603.46
3 id * time -2253.21 0.16 -0.065,0.83 1.24112 0.17 4 4514.43 4532.40
4 (group + id) * time -2231.13 0.13 0.102,0.058,0.745 1.59394 0.18 5 4472.27 4494.73
5 group * id * time -2232.78 0.18 0.11,0.058,0.751 1.19639 0.18 5 4475.55 4498.01

Outro

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Additive interaction modelling using I-priors

github.com/haziqj/iprior-interaction

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kernel regression rkhs hilbert-spaces fisher-information rkks empirical-bayes krein-spaces reproducing-kernel

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